Question

If two numbers $p$ and $q$ are chosen randomly from the set $\{1, 2, 3, 4\}$, one by one, with replacement, then the probability of getting $p^2 \ge 4q$ is

• A. $\frac{1}{4}$
• B. $\frac{7}{16}$
• C. $\frac{1}{2}$
• D. $\frac{9}{16}$

Hint:

For small sets, listing cases systematically by fixing one variable ($p$) is the safest way to avoid mistakes.

Answer 1

The Correct Option is B

Step 1: Concept
The total number of possible pairs $(p, q)$ is $4 \times 4 = 16$ because drawing is done with replacement.

Step 2: Meaning
List the pairs that satisfy the condition $p^2 \ge 4q$.

Step 3: Analysis
- If $p=1: 1^2 \ge 4q \implies 1 \ge 4q$ (No $q$ values). - If $p=2: 2^2 \ge 4q \implies 4 \ge 4q \implies 1 \ge q$ (Pair: (2, 1)). - If $p=3: 3^2 \ge 4q \implies 9 \ge 4q \implies 2.25 \ge q$ (Pairs: (3, 1), (3, 2)). - If $p=4: 4^2 \ge 4q \implies 16 \ge 4q \implies 4 \ge q$ (Pairs: (4, 1), (4, 2), (4, 3), (4, 4)). Total favorable cases $= 1 + 2 + 4 = 7$.

Step 4: Conclusion
Probability $= \frac{\text{Favorable cases}}{\text{Total cases}} = \frac{7}{16}$. Final Answer: (B)